ABSTRACT: We describe an infinitary logic for metric structures which is
analogous to $L_{\omega_1, \omega}$. Using topological methods, we prove
an omitting types theorem for countable fragments of this logic. We use
omitting types, together with an analogue of Scott’s theorem, to show that
every non-trivial separable quotient of a non-separable Banach space is
almost isometric to a quotient of a Banach space of density $\aleph_1$.